Geodesic
A new characterization of the Poisson distribution
Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 879-882 
V. M. Kruglov. A new characterization of the Poisson distribution. Matematičeskie zametki, Tome 20 (1976) no. 6, pp. 879-882. http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a9/
@article{MZM_1976_20_6_a9,
     author = {V. M. Kruglov},
     title = {A~new characterization of the {Poisson} distribution},
     journal = {Matemati\v{c}eskie zametki},
     pages = {879--882},
     year = {1976},
     volume = {20},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_6_a9/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this note we show that an infinitely divisible (i.d.) distribution function $F$ is Poisson if and only if it satisfies the conditions $F(+0)>0$, for any $0<\varepsilon<1$ $$ \int_{-\infty}^{1-\varepsilon}\frac{|x|}{1+|x|}\,dF=0, $$ and for any $0<\alpha<1$ $$ \int_0^\infty e^{\alpha x\ln(x+1)}\,dF<\infty $$